Systematic study of quantum groups from the viewpoint of operator algebras

从算子代数角度系统研究量子群

• 批准号: 12640199
• 负责人: YAMANOUCHI Takehiko
• 金额: $ 2.3万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640199/
• 关键词: Locally compact quantum group; von Neumann algebra; Cartan subalgebra; Galois group; Nontype I C^*-algebra; Approximately inner automorphism; Amalgamated free product; Jones index; 融合積; 作用素環; 量子群; 極小作用; UHF flow; 純粋状態空間; 基底状態; 自由積; カッツ環

Yamanouchi has made an intensive study of actions of locally compact quantum groups on von Neumann algebras. He introduced a notion of a Galois group for such an action. He has succeeded in topologically identifying the Galois group of an intergrable, minimal action of a general locally compact quantum group as the intrinsic group of the dual quantum group.Kishimoto has made a very unique research on automorphisms or one-parameter automorphism groups on nontype I separable C^*-algebras. He showed that, for any irreducible representation of such a C^*-algebra without nonzero intersection with compact operators, there exists an approximately inner one-parameter automorphism group so that the representation is covariant with respect to this one-parameter automorphism group.Arai has made a progress in his research in mathematical physics. He considered a model of quantum particles coupled to a massless quantum scalar field, and showed that the model has a ground state for all values of the coupling constant.Ueda has made an anlysis on amalgamated free products of von Neumann algebras over Cartan subalgebras. By utilizing an irreducible inclusion of type III factors coming from his free-product type action of the quantum SU_q(2), he has been ble to solve the long-standing problem that the free group factor of Radulescu possesses irreducible subfactors of arbitrary index greater than 4.

Yamanouchi深入研究了局部紧量子群在冯诺依曼代数上的作用。他介绍了一个概念的伽罗瓦集团这样一个行动。他成功地在拓扑上将一般局部紧量子群的可积极小作用的伽罗瓦群确定为对偶量子群的内禀群;岸本对非I型可分C^*-代数上的自同构或单参数自同构群进行了非常独特的研究。他证明了，对于这样一个与紧算子没有非零交集的C^*-代数的任何不可约表示，存在一个近似内单参数自同构群，使得该表示关于这个单参数自同构群是协变的。他考虑了一个量子粒子与无质量量子标量场耦合的模型，并证明了该模型对所有的耦合常数都有基态。上田对Cartan子代数上vonNeumann代数的合并自由积作了分析。利用他对量子SU_q（2）的自由乘积型作用量所产生的III型因子的不可约包含，他解决了长期存在的问题，即Radulescu的自由群因子具有任意指数大于4的不可约子因子.

项目成果

Akitaka KISHIMOTO: "Quasi-product flows on a C^*-algebra"Communications in Mathematical Physics. 229. 397-413 (2002)

Akitaka KISHIMOTO：“C^*-代数上的准积流”数学物理通讯。

Takehiko Yamanouchi: "Uniqueness of Haar measures for a quasi Woronowicz algebra"Hokkaido Mathematical Journal. 30. 105-112 (2001)

Takehiko Yamanouchi：“拟 Woronowicz 代数的 Haar 测度的唯一性”北海道数学杂志。

Asao Arai: "Exact solutions of multi-component nonlinear Schrodinger and Klein-Gordon equations in two-dimensional space-time"J. Phys. A : Math. Gen.. 34. 4281-4288 (2001)

Asao Arai：“二维时空中多分量非线性薛定谔和克莱因-戈登方程的精确解”J。

Takehiko YAMANOUCHI: "Galois groups of quantum group actions and regularity of fixed-point algebras"Transactions of the American Mathematical Society. (掲載予定).

Takehiko YAMANOUCHI：“量子群作用的伽罗瓦群和定点代数的正则性”美国数学会汇刊（即将出版）。

Takehiko YAMANOUCHI: "Description of the automorphism group Aut(A/A^∧{alpha}) for a minimal action of a compact Kac algebra and its application"Journal of Functional Analysis. 194. 1-16 (2002)

Takehiko YAMANOUCHI：“紧致 Kac 代数最小作用的自同构群 Aut(A/A^∧{alpha}) 的描述及其应用”泛函分析杂志 194. 1-16 (2002)。